Stefano Martin

Relative generalized Hamming weights of one-point algebraic geometric codes

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes [23], [22]. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights.

An improvement of the Feng---Rao bound for primary codes

We present a new bound for the minimum distance of a general primary linear code. For affine variety codes defined from generalised

On asymptotically good ramp secret sharing schemes

C_{ab}

Relative generalized Hamming weights of

Cab polynomials the new bound often improves dramatically on the Feng–Rao bound for primary codes (Andersen and Geil, Finite Fields Appl 14(1):92–123, 2008; Geil et al., Lecture Notes in Computer Science 3857: 295–306, 2006). The method does not only work for the minimum distance but can be applied to any generalised Hamming weight.

q

Salazar, Dunn and Graham in [16] presented an improved Feng-Rao bound for the minimum distance of dual codes. In this work we take the improvement a step further. Both the original bound by Salazar et al., as well as our improvement are lifted so that they deal with generalized Hamming weights. We also demonstrate the advantage of working with one-way well-behaving pairs rather than weakly well-behaving or well-behaving pairs.

-ary Reed-Muller codes

We coin the term of asymptotically good sequences of ramp secret sharing schemes. These are sequences such that when the number of participants goes to infinity, the information rate approaches some fixed positive number while the worst case information leakage of a fixed fraction of information, relative to the number of participants, attains another fixed number. A third fixed positive number describes the ratio of participants that are guaranteed to be able to recover all – or almost all – of the secret. By a non-constructive proof we demonstrate the existence of asymptotically good sequences of schemes with parameters arbitrarily close to the optimal ones. Moreover, we demonstrate how to concretely construct asymptotically good sequences of schemes from sequences of algebraic geometric codes related to a tower of function fields. Our study involves a detailed treatment of the relative generalized Hamming weights of the involved codes.

Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower

We propose a new method for constructing small-bias spaces through a combination of Hermitian codes. For a class of parameters our multisets are much faster to construct than what can be achieved by use of the traditional algebraic geometric code construction. So, if speed is important, our construction is competitive with all other known constructions in that region. And if speed is not a matter of interest the small-bias spaces of the present paper still perform better than the ones related to norm-trace codes reported in [12].